a. X-ray and Neutron Compound Refractive Lenses
X-rays and neutrons can be collected, collimated, and focused using a series of small-aperture, thin, biconcave lenses with a common optical axis. M. A. Piestrup, J. T. Cremer, R. H. Pantell and H. R. Beguiristain (U.S. Pat. No. 6,269,145 B1, which is incorporated herein by reference), teach that an stack of individual thin unit lenses 12 without a common substrate, but with a common optical axis 10, forms a compound refractive x-ray lens 14, which is capable of collecting and focusing x-rays in a short focal length (as shown in FIG. 1). X-rays and neutrons 45 are focused by the compound refractive lens 14 along an optical axis 10 to a focal point 16. The closely spaced series of Nx bi-concave unit lenses 12 each of focal length f1, result in a focal length f of:                     f        =                                            f              1                                      N              x                                =                                    R                              2                ⁢                N                ⁢                                                                  ⁢                δ                                      .                                              (        1        )            The unit lens focal length f1 is given by:                                           f            1                    =                      R                          2              ⁢              δ                                      ,                            (        2        )            where the complex refractive index of the unit lens material is expressed by:                               n          =                      1            -            δ            +                                          i                ⁡                                  (                                      λ                                          4                      ⁢                      π                                                        )                                            ⁢              μ                                      ,                            (        3        )            R is the radius of curvature of the lens, λ is the neutron or x-ray wavelength and μ is the linear attenuation coefficient of the lens material. For cylindrical unit lenses R=Rh, the radius of the cylinder; for spherical lenses R=Rs, the radius of the sphere; for the case of parabolic unit lenses R=Rp, the radius of curvature at the vertex of the paraboloid.
Equation (1) shows that the total focal length has been reduced by 1/Nx. The focal length of a single unit lens 12 would be extremely long (e.g. 100 meters), but using 100 of such unit lenses 12 would result in a focal length of only 1 meter. This makes the focusing, collecting and imaging of objects with x-rays and neutrons possible with much shorter focal lengths than was thought possible.
Unfortunately, the aperture of the compound refractive lens is limited. This is due to increased absorption at the edges of the lens as the lens shape may be approximated by a paraboloid of revolution that increases thickness in relation to the square of the distance from the lens axis. These effects make the compound refractive lens act like an iris as well as a lens. For a radius R=Rh, Rs, or Rp, the absorption aperture radius ra is:                               r          a                =                                            (                                                2                  ⁢                  R                                                  μ                  ⁢                                                                          ⁢                                      N                    x                                                              )                                      1              2                                =                                                    (                                                      4                    ⁢                    δ                    ⁢                                                                                  ⁢                    f                                    μ                                )                                            1                2                                      .                                              (        4        )            
If the lenses refract with spherical surfaces, only the central region of the lens approximates the required paraboloid-of-revolution shape of an ideal lens. The parabolic aperture radius rp, where there is a π phase change from the phase of an ideal paraboloid of revolution, is given by:                                           r            p                    =                                    2              ⁢                                                (                                                                                    (                                                                              N                            x                                                    ⁢                          f                          ⁢                                                                                                          ⁢                          δ                                                )                                            2                                        ⁢                    λ                    ⁢                                                                                  ⁢                                          r                      i                                                        )                                                  1                  4                                                      ≈                          2              ⁢                                                (                                                                                    (                                                                              N                            x                                                    ⁢                          δ                                                )                                            2                                        ⁢                                          f                      3                                        ⁢                    λ                                    )                                                  1                  4                                                                    ,                            (        5        )            where r1 is the image distance and λ is the x-ray wavelength. Rays outside this aperture do not focus at the same point as those inside. The approximation in (5) is true for a source placed at a distance much larger than f.
For imaging, the effective aperture radius re is the minimum of the absorption aperture radius, ra, the parabolic aperture radius, rp, and the mechanical aperture radius rm; that is:re=MIN(ra,rp,rh).  (6)However, since lens shape can be made parabolic and the mechanical aperture can be made larger, the absorption aperture ra is usually the limiting aperture. For example, using Beryllium as a lens material for x-rays, the absorption aperture is below 1 mm in diameter for x-rays. For cold neutrons the Be lenses are bigger (e.g. 2–4 cm diameter), but the sources are even larger, requiring even larger apertures. Thus the compound refractive lens' apertures are small and limited in their ability to capture the total image or collect most of the flux from sources of neutrons or x-rays.
Since one can always make the mechanical aperture of a lens bigger and, in most cases make the lenses parabolic, the absorption aperture is the dominant determining parameter of the compound refractive lenses aperture size. Note from equation 4, if desire shorter focal lengths f, then the absorption apertures get smaller (e.g. If Kapton is used as the lens material, the absorption aperture for a compound refractive lens is only 2ra=100 μm for x-ray photon energies of around 8 keV).
Compound refractive lenses for neutrons and x-rays have been made using a variety of techniques. For focusing and imaging the lenses need to be either bi-concave or plano-concave. They can also be Fresnel lenses with the additional requirement that individual zones need to be aligned accurately as described in U.S. Pat. No. 6,269,145, by M. A. Piestrup et al. The x-ray lenses have been made using compression molding for 2-D lenses (U.S. Pat. No. 6,269,145 B1 May 1998, M. A. Piestrup, R. H. Pantell, J. T. Cremer and H. R. Beguiristain, “Compound Refractive Lens for X-rays,”) and drilling for 1-D lenses (U.S. Pat. No. 5,594,773, Toshihisa Tomie, “X-ray Lens”). Bi-concave lenses have been formed by using a capillary filled by epoxy and filled with a series of bubbles: the interface between two bubbles forms a bi-concave lens and a series of such bubbles forms a multi-lens path down the axis of the capillary (Yu. I Dudchik, N. N. Kolchevsky, “A microcapillary lens for X-rays, Nuclear Instruments and Methods A421, 361 (1999)).
b. Visible Optics Arrays of Microlenses
Planar (2-Dimensional, 2-D) optical arrays of microlenses have been used to produce short focal length imaging systems for visible electromagnetic radiation. U.S. Pat. No. Re. 28,162 by R. H. Anderson entitled “Optical Apparatus Including a Pair of Mosaics of Optical Imaging Elements,” describes an apparatus which can be used as an image transmission system or a part of a camera's optics for photographing the trace produced on the fluorescent screen of a cathode ray oscilloscope. An optical apparatus is described, which includes two 2-D (or planar) arrays of microlenses forming a plurality of light paths each containing image inverting and erecting elements in different planar arrays (or mosaics), which transmit different portions of an image and recombine such image portions with their original object. A plurality of aperture plates is used to prevent undesired light from reaching the composite image formed on the final image surface, and the lens pairs are spaced so that adjacent image portions partially overlap to provide a single final image.